In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
, Helmholtz's theorems, named after
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
, describe the three-dimensional motion of fluid in the vicinity of
vortex lines. These theorems apply to
inviscid flows and flows where the influence of
viscous forces are small and can be ignored.
Helmholtz's three theorems are as follows:
;Helmholtz's first theorem:
:The strength of a vortex line is constant along its length.
;Helmholtz's second theorem:
:A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.
;Helmholtz's third theorem:
:A fluid element that is initially irrotational remains irrotational.
Helmholtz's theorems apply to inviscid flows. In observations of vortices in real fluids the strength of the vortices always decays gradually due to the dissipative effect of
viscous forces.
Alternative expressions of the three theorems are as follows:
# The strength of a vortex tube does not vary with time.
# Fluid elements lying on a vortex line at some instant continue to lie on that vortex line. More simply, vortex lines move with the fluid. Also vortex lines and tubes must appear as a closed loop, extend to infinity or start/end at solid boundaries.
# Fluid elements initially free of
vorticity remain free of vorticity.
Helmholtz's theorems have application in understanding:
*
Generation of lift on an
airfoil
An airfoil (American English) or aerofoil (British English) is a streamlined body that is capable of generating significantly more Lift (force), lift than Drag (physics), drag. Wings, sails and propeller blades are examples of airfoils. Foil (fl ...
*
Starting vortex
*
Horseshoe vortex
*
Wingtip vortices
Wingtip vortices are circular patterns of rotating air left behind a wing as it generates Lift (force), lift. The name is a misnomer because the cores of the vortex, vortices are slightly inboard of the wing tips. Wingtip vortices are sometimes ...
.
Helmholtz's theorems are now generally proven with reference to
Kelvin's circulation theorem. However Helmholtz's theorems were published in 1858,
nine years before the 1867 publication of Kelvin's theorem.
Notes
References
* M. J. Lighthill, ''An Informal Introduction to Theoretical Fluid Mechanics'', Oxford University Press, 1986,
*
P. G. Saffman, ''Vortex Dynamics'', Cambridge University Press, 1995,
*
G. K. Batchelor, ''An Introduction to Fluid Dynamics'', Cambridge University Press (1967, reprinted in 2000).
* Kundu, P and Cohen, I, ''Fluid Mechanics'', 2nd edition, Academic Press 2002.
* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 4th edition, Academic Press: San Diego (1995) pp. 92–93
* A.M. Kuethe and J.D. Schetzer (1959), ''Foundations of Aerodynamics'', 2nd edition. John Wiley & Sons, Inc. New York
{{DEFAULTSORT:Helmholtz's Theorems
Aerodynamics
Vortices
Theorems in mathematical physics
Hermann von Helmholtz